On the upper bound of number-theoretic functionf F_f(h)

Shenghan Lu The high school attached to Tsinghua University

S.-T. Yau High School Science Awarded Papers mathscidoc:1608.35027

The problem of the existence of infinitely many prime values of a number-theoretic function f (x) has been one of the most important topics in Number Theory. Note that if f (x) represents infinitely many primes, then we can get this necessary condition: for any positive integer h , there exists a positive integer k such that ( f (k), h) =1and f (k) >1. Naturally, we are interested in the number-theoretic functions f (x) that satisfy the aforementioned necessary condition. Thus, there must exist the least positive integer n such that (f(n),h)-1andf(k)>1. Denote this least positive integer n byF_f(h). In this paper, we mainly focus on three famous number-theoretic functions:&f(x)=2^{2^x}+1,m(x)=2^x-1 and l(x)=x^2+1& proving they satisfy the aforementioned necessary condition respectively. Furthermore, we approximately estimate the upper bound of F_f(x)(h),Fm(x)(h),Fl(x)(h) respectively, and obtain some interesting results.
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  title={On the upper bound of number-theoretic functionf F_f(h)},
  author={Shenghan Lu},
Shenghan Lu. On the upper bound of number-theoretic functionf F_f(h). 2009. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160813215155201274075.
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